Wavelet functional principal component analysis for batch process monitoring

To facilitate the understanding and analysis of process conditions, a novel wavelet functional principal component analysis is proposed for monitoring batch processes from the functional perspective. In the proposed method, the variables' trajectories are taken as smooth functions instead of discrete vectors. To this end, the original discrete variables are transferred into continuous functions using wavelet basis functions in an active way. This can not only highlight the subtle shape differences between the normal and faulty variables trajectories but also easily address the uneven-length issue in practical batch processes. Additionally, without unfolding the operation, the 3D matrix is transferred into the functional matrix directly. The functional principal component analysis method is then performed on the functional space to establish monitoring models. Thanks to the compact-support characteristics of the wavelet functions, the proposed method can be directly applied to within-batch detection without data pre-treatment. A numerical case, a case of the simulated penicillin fermentation process, and a case of the laboratorial injection molding process are given to demonstrate the effectiveness of the proposed method.